Multiscale Analysis and Numerical Modeling of the Portevin-Le Chatelier Effect

Transcription

1 International Journal for Multiscale Computational Engineering, 3(2) (2005) Multiscale Analysis and Numerical Modeling of the Portevin-Le Chatelier Effect Zhongjia Chen CAS Key Laboratory of Mechanical Behavior and Design of Materials University of Science and Technology of China Hefei , China Department of Materials Science and Engineering, Hefei University of Technology Hefei , China Qingchuan Zhang* & Xiaoping Wu CAS Key Laboratory of Mechanical Behavior and Design of Materials University of Science and Technology of China Hefei , China ABSTRACT The Portevin-Le Chatelier (PLC) effect refers to one type of plastic instability, which often manifests itself as discontinuous yielding and localized deformation in some metallic alloys deformed under certain conditions. A phenomenological model based on a multiscale analysis is developed to investigate the PLC effect. In this model, a new component of stress is introduced, which takes account of the collective interactions between mobile dislocations and solute atoms, to describe the influence of dynamic strain aging (DSA) on the flow stress. The effects of microscopic pinning and unpinning of dislocations on the macroscopic deformation behavior are considered in an integrative and competitive manner. Due to the competition of these two effects during deformation, the alloys may exhibit the negative strain rate sensitivity of flow stress, which is a necessary condition for the occurrence of the PLC effect. A nonuniform spatial distribution of some material parameters was used in the model to reflect the heterogeneous nature of the deformed material, including a linear change of the initial cross-sectional area and a random perturbation of the initial internal stress. Numerical simulations based on this heterogeneous model were carried out for tensile testing of aluminum alloy 2017, by which the serrated yielding and localized deformation behavior were successfully reproduced. The results also indicate the relation between the macroscopic jerky flow and the pinning/unpinning of dislocations at the micro level. Address all correspondence to Qingchuan Zhang, zhangqc@ustc.edu.cn /04/$35.00 c 2005 by Begell House, Inc. 227

2 228 CHEN ET AL 1. INTRODUCTION For some metallic alloys deformed in certain ranges of temperature and strain rate, the Portevin-Le Chatelier (PLC) effect, or so named serrated yielding, is a spectacular phenomenon often observed after an initially smooth deformation. Typical of this effect is a complex behavior both in time and in space. The temporal dynamics exhibits itself by repeated stress serrations on stress-strain curves, and thus the subsequent deformation takes place in a discontinuous and jerky manner. Corresponding to each stress serration, there is a localized deformation band (the PLC band) being initiated and propagating along the specimen, which spatially depicts the nonuniform landscape of the PLC effect. Recently, much renewed attention has been paid to the investigation of the PLC effect toward a comprehensive understanding of this phenomenon, owing to both theoretical and applicative interests. The localized deformation behavior associated with this particular plastic instability will, generally speaking, reduce the total fracture elongation of the deformed material [1]. Since the PLC band is actually a traveling neck in which a very large shear deformation has been severely concentrated, its capacity for further deformation is exhausted, and the final crack has always been found to be located at the place where the localized shear band stops [2,3]. This kind of loss of plasticity induced by localized deformation is also common in nanocrystalline materials [4,5], although the underlying mechanism and the intrinsic length scale are not exactly the same for these two situations. Another drawback involved in the PLC effect is that some undesirable striations remain on the surfaces of the deformed material [6,7]. These banded deformation markings, visible even with naked eye due to the macroscopically localized shearing, no doubt plague the surface quality, especially for sheet products. It has been experimentally proven that many dilute alloys suffer from the PLC effect at their generally used temperatures and strain rates, e.g., for Al alloys it is most prominent at room temperature, which causes difficulty in their forming and shaping and further limits their use, despite their envied advantage of high strength-to-weight ratio. Therefore, from the application point of view, besides some critical conditions for the PLC effect to occur, how to control this plastic instability and the associated localized deformation is of great importance and the issue of most concern for engineering. To get a proper answer to the above question, one must see deep into the fundamental mechanisms and the related structure evolutions on different length scales. The physical basis, well accepted so far, explains the PLC effect as the result of interaction between mobile dislocations and solute atoms in the dynamic strain aging (DSA) regime [8 13] since this phenomenon does not appear in pure metals. Based on this concept, some time-dependent models have been established that can give a satisfactory description of some temporal features of the PLC effect (see Ref. [14] for a review). However, the research by Hahner et al. [15,16] shows that DSA is not the only condition for PLC instability to occur; but in addition, cooperative dislocation interactions are necessary. Recently, some developments have been made in the investigation of the spatial characteristics of the PLC band [17 19], i.e., the static, hopping, and propagating shear bands are captured by embedding the local constitutive equations into the finite element (FE) simulations. Compared with the rich experimental observations [2,3] obtained by using dynamic digital speckle pattern interferometry, some other dramatic spatial aspects of this phenomenon are still not fully resolved, e.g., two bands coexist simultaneously; the band inclination angle may change symmetrically, sometimes in the later half of International Journal for Multiscale Computational Engineering

3 MULTISCALE ANALYSIS AND NUMERICAL MODELING 229 tests; and shrinkage deformation takes place outside the band, even in tensile tests. Since the macroscopic plastic deformation is governed by the mechanical properties of microstructures and their interactions on lower length scales, one conceptual way to achieve a thorough understanding of the physics and process controlling the PLC effect is to put it in the scheme of multiscale modeling, namely, the study could be taken from atomistic calculations of the dislocation-solute interactions and the dislocation structures, to examination of the related spatiotemporal dynamics at the macro level. Multiscale modeling has become a catchphrase in materials science during the last decade, and some significant advances have been made (see Refs. [20,21] for reviews and the references therein). However, at the present state of the art, the atomic-to-continuum approach, no matter if it is of the hierarchical (information passing) or hybrid (coupling) methodology, can be practicable only for some particular problems in which the critical length scale is relatively small, such as a crack tip, grain boundary, or nanoindention. The recent research [22,23] based on dislocation dynamics made a distinct step forward in the multiscale modeling of shear localization, but again, this method can only deal with small-scale plasticity phenomena. Therefore, approaches from meso (grain-length) to the macro level should presently be most tractable for multiscale modeling of the PLC effect and the localized deformation band involved in it, since the related length scale here is widely ranged from atomic to macroscopic millimeters. Recent attempts [18,19], in which crystal plasticity is coupled with a three-dimensional FE framework, can be attributed to this category. When performing such work, a local constitutive model must be preliminarily well defined, with some basic or phenomenological understanding of the underlying mechanisms and the response of microstructures. In this paper, the PLC effect is studied based on a multiscale analysis, and then the interactions between mobile dislocations and solute atoms are reexamined in a collective manner. With the emphasis on the macroscopic mechanical response of the interaction and evolution of microstructures, a phenomenological model of the PLC effect is developed. To verify the applicability of this model, simple one-dimensional numerical simulations were carried out for tensile tests of Al alloy 2017 specimens at room temperature. 2. CONTINUUM MODEL BASED ON MULTISCALE ANALYSIS Dislocation is the carrier of plastic deformation, and the macroscopic strength of the deformed material depends on the resistance to the dislocation motion at lower length scales. The dislocation multiplication and varieties of deformation-induced dislocation structures will increase the obstacles and their intensities to be surmounted by the dislocation motion; in other words, additional force is needed for further plastic deformation. Making this strainhardening effect clear has been one conundrum of the materials science field for a long time, ever since the dislocation theory was established. In this paper, a simple power rule is assumed as follows to characterize the response of flow stress to long-range dislocation interactions, since the strain hardening coefficient shows a decreasing tendency with the increasing plastic strain ε on the testing stress-strain curves. σ ε = h ε m (1) where the coefficients h, m are considered as constants for convenience, and m <1. Besides the long-range obstacles, there are also some short-range impediments in the dislocation slip path. The latter can be conquered by the thermally activated dislocation motion Volume 3, Number 2, 2005

4 230 CHEN ET AL with the aid of effective stress acting on the dislocation line, and this gives an explanation for the rate dependence of plastic flow. The strain rate dependence of flow stress is described here in two terms. One comes out as an instantaneous stress response that expresses the normal positive strain rate sensitivity. Taking account of the Orowan equation and the Arrhenius equation, where only the linear stress term is retained in the expression for the activation energy [10], the instantaneous contribution of the plastic strain rate to flow stress is given by σ ε = f ln ( ε / ε ) (2) where ε is a sufficiently small constant, or in its differential form dσ ε = f d ln ε (3) In Eqs. (2) and (3), f is the instantaneous strain rate sensitivity and f > 0. The other dependence of flow stress on strain rate, i.e., the transient response, is included in a new stress component, σ dsa, to describe the collective interactions between mobile dislocations and solute atoms due to dynamic strain aging. Since a dislocation is a disruption of the stacking sequence of atomic planes in a crystal, atoms near the dislocation line do not occupy the lowest energy sites, and high stresses will be found in the vicinity of the dislocation. For alloys, a solute atom also distorts its surrounding lattice on account of the solute and solvent atoms possessing different atomic sizes. Hence, there is an energy well at the dislocation for solutes around it, viz., and these impurities may diffuse toward the dislocation in its elastic stress field to minimize the system energy. A Cottrell atmosphere of solute atoms is formed as a result of this stress-induced redistribution of solutes in the vicinity of the dislocation, which may work as an impediment to dislocation motion. The force acting on a dislocation must be sufficiently increased to overcome the interaction between the dislocation and its solute cloud before it is about to move. Considering the discontinuity of dislocation motion, a dislocation can be captured and locked again by solute atoms during its waiting time at obstacles in the slip path when the solute mobility becomes comparable to that of dislocation. Therefore, this pinning and unpinning process of dislocation movement may take place repeatedly in the regime of dynamic strain aging, and the stress oscillations result correspondingly. It is difficult, at present, to quantify the kinetics of dynamic strain aging on the microscopic scale because the associated physical principle is very complicated and even enigmatic, and this is also not the main purpose of the present paper. With the emphasis on macroscopic mechanical response of dynamic strain aging, the following semiempirical expression [24,25] is adopted here to characterize the stress component resulting from collective interactions between mobile dislocations and solute atoms dσ dsa /dt = a ε σ dsa + b exp[(σ M dsa σ dsa)/g] 1 (4) where a, b, g are positive constants, and σdsa M is the maximum value of σ dsa. Breakaway of a dislocation from its Cottrell atmosphere will cause a decrease of σ dsa, which is reflected in the first term on the right-hand side of Eq. (4). The farther the dislocation is pulled away from its atmosphere, or on the macroscopic scale, the bigger the plastic strain increment and the stronger the original resistance, the more the reduction of this stress component. The second term takes account of the age hardening effect, which is correlated to solute diffusion and, accordingly, is time dependent. The concentration of solute atoms at the dislocation, C s, increases with time, but there should eventually exist a saturation value, C m, for it since the International Journal for Multiscale Computational Engineering

5 MULTISCALE ANALYSIS AND NUMERICAL MODELING 231 diffusion velocity will decrease with increasing C s. Correspondingly, σ dsa develops to its maximum value, σdsa M, at a decreasing rate, which is assumed in an exponential form as shown in Eq. (4). In other words, dynamic strain aging is considered in this paper as the combination of two effects, the age hardening and the subsequent strain-induced softening. The mechanical behavior of the alloy should be related to the competition between these two effects. For example, if the softening effect is dominant, when the absolute value of the first term in the righthand part of Eq. (4) is larger than that of the second term, a sudden load drop would be expected in tensile testing. On the contrary, when the age hardening effect is ascending, the material should be in the period of reloading. In a steady state (strain rate is kept constant), which means these two effects are in balance, one can get or dσ dsa /dt = 0 (5) a ε σ dsa = b exp[(σ M dsa σ dsa)/g] 1 (6) It is apparent from Eq. (6) that a larger ε will result in a smaller σ dsa, so in this model the negative strain rate sensitivity of flow stress necessary for the occurrence of the PLC effect [26] has its definite physical basis, and then can be satisfied naturally [27] rather than being imposed as a presupposition. Using an initial internal stress σ 0, the total flow stress becomes σ = σ 0 + σ ε + σ ε + σ dsa (7) and its incremental form can be written as dσ = (m hε m 1 a σ dsa )dε + f d ln ε + b{exp[(σ M dsa σ dsa)/g] 1 }dt (8) 3. NUMERICAL SIMULATION OF TENSILE TESTS 3.1 Initial Conditions Numerical simulations based on the above model were carried out for uniaxial tension at room temperature of an aluminum alloy 2017 specimen. In order to get the initial values of ε and σ dsa in the simulations, the following equation: P = K(vt l 0 ε) + P 0 (9) is used for the load P, in which K is the effective stiffness of the machine and specimen, v is the tensile speed, l 0 is the initial specimen gauge length, and P 0 is a constant at the initial values t 0 = 0 and ε 0 = 0 (initial plastic strain). Considering the decrease of the initial cross-sectional area S 0 during plastic deformation, the stress is expressed as σ = P (1 + ε)/s 0 = [K(vt l 0 ε) + P 0 ](1 + ε)/s 0 (10) The initial plastic strain rate ε 0 is considered as the homogeneous deformation rate, so combining Eq. (10) with Hooke s law, it can be derived as ε 0 = K v/ (K l 0 + E S 0 ) (11) where E stands for Young s modulus. The initial value σ dsa0 can be calculated by substituting ε 0 into the steady-state expression of Eq. (6) a ε 0 σ dsa0 = b{exp[(σ M dsa σ dsa0)/g] 1 } (12) 3.2 Consideration of the Heterogeneity of Plastic Deformation Plastic deformation is inhomogeneous on the microscopic scale owing to the microstructural Volume 3, Number 2, 2005

6 232 CHEN ET AL heterogeneity of the deformed material. The PLC effect actually reflects the macroscopic response of the alloy to the evolution of its microscopic structure under specific deformation conditions (including the specimen geometry). It must be noted that this microscopic nonuniformity has developed to its extreme state in the PLC effect; namely, it has become the prominent deformation mode even on the macroscopic scale. This peculiarity, therefore, must be considered appropriately in the numerical simulations of the phenomenon. In order to spatially model heterogeneous deformation, the specimen gauge length was numerically divided into N sections of equal width, simply coupled with one another through the acting force; and there must be a nonuniform distribution of the model parameters along the specimen to reflect, on the macroscopic scale, the intrinsic heterogeneity of the deformed material. But each section is assumed to have the same parameters, for convenience of computation in the numerical simulation. So, the deformation in each section will be uniform, or furthermore, the individual section is the unit of localized deformation. Two different types of defect were introduced in the numerical simulation: a linear change of the initial crosssectional area with the total amplitude of 1%S 0 to describe the possible geometric defect in the specimen, and a random perturbation of the initial internal stress σ 0 with the total amplitude of 5 MPa to reflect the heterogeneity of the microstructure of the specimen. In the numerical simulation, these two types of defect were introduced once the plastic deformation was started. In other words, they take effect only after the plastic deformation begins and denote the difference of deformation resistance among the specimen sections. During succedent calculations, the plastic deformation was dissimilar in each section, but the evolution of this dissimilarity was dominated by the established constitutive relationships. As it was stated above that each section can be regarded as the element of the macroscopically localized plastic deformation when the initial disturbances are prescribed in the numerical simulations, the number of the discretized sections N is determined here according to the result that the width of the PLC band is approximate to the specimen thickness [2,3,17]. 3.3 Results and Discussion The model parameters used in the simulations are given as: h = 350 MPa, m = 0.45, f = 1 MPa, ε =10 8 /s, a = 800, b = 0.08 MPa/s, g = 2.5 MPa, σ M dsa = 15 MPa, σ 0 = 25 MPa, E = 57 Gpa. To compare with the simulation results, experimental tensile tests of 2017 flat specimens were also carried out at room temperature. The parameters for the testing system are K = N/mm, l 0 = 50 mm, and S 0 = 60 mm 2. Also, the specimen was equally divided into 20 sections in the simulations, N = 20, on account of its 3 mm thickness. The experimental and modeling serrated load curves are shown in Figs. 1 4 with the imposed (nominal) strain rates of s 1 and s 1, respectively. The simulation results FIGURE 1. Experimental results at the imposed strain rate of s 1 International Journal for Multiscale Computational Engineering

7 MULTISCALE ANALYSIS AND NUMERICAL MODELING 233 FIGURE 2. Simulation results at the imposed strain rate of s 1 obtained by assuming a linear variation of the initial cross-sectional area FIGURE 4. Simulation results at the imposed strain rate of s 1 obtained by assuming a random perturbation of the initial internal stress FIGURE 3. Experimental results at the imposed strain rate of s 1 in Figs. 2 and 4 were obtained by introducing the perturbation of the initial cross-sectional area and initial internal stress, respectively. It is proved that the serrated yielding phenomenon can be successfully reproduced from the nonuniformity of both geometry and inner structure, but the latter reflects the physical heterogeneous nature of the deformed material, and is thus more constitutive. Comparing with our earlier results [25,28], it can be seen that a disturbance is needed here merely to trigger the PLC instability and its detailed configuration is not the controlling factor, viz., the disturbance itself rather than its distribution function is an inherent point of the model. The similar result to clarify the operation of the prescribed initial nonuniformity was also achieved in the research [29] about the deformation behavior of an amorphous polymer, where a normal distribution of the initial shear strength of the polymer was employed to indicate the heterogeneous distribution of the molecular chains. The localized deformation behavior is also investigated here. Figures 1 4 also show the experimental and simulated strain-time curves. The experimental results shown in Figs. 1 and 3 were measured with an extensometer of 25 mm gauge length, which was placed in the middle of the specimen during test. Apparently, the curve is not smooth even though it figures the average strain of the specimen between the two edges of the extensometer, and this adequately indicates the localized deformation property in the specimen. The position where the crack occurred was located inside the extensometer gauge length, so the slope be- Volume 3, Number 2, 2005

8 234 CHEN ET AL comes higher at the end of the curve. Comparing with this result, the nonuniform and localized deformation behavior is also reproduced theoretically through the computational strain curves displayed in Figs. 2 and 4. These calculated strain data were obtained by averaging the values of 10 sections that fell within the extensometer gauge length. Besides the average deformation behavior, numerical simulation can also give an insight into the details of the jerky flow. Figure 5 shows the simulated strain of three adjacent sections, calculated by introducing the perturbation of initial internal stress. The strain versus time curve of each section initially has the same slope that is approximate to the imposed strain rate, when there is no serration in the calculated load curve. This indicates uniform elongation in the specimen. However, such balance is disturbed by the succeeding localized deformation. It is evident that each curve consists of repeated strain plateaus and sharply increasing steps that correspond to sudden stress drops and denote avalanchelike shear deformation in this section. The strain steps become higher and wider with increasing plastic deformation and, meanwhile, the stress serrations exhibit the same regularity. This means that the PLC bands propagate repeatedly along the specimen with a decreasing velocity, as was directly observed in Refs. [2,3]. From strain-time curves, the propagating characteristic of the PLC bands can be captured as bands of type A [6], clearly depicted in Fig. 5 propagating at an initial velocity of about 2 mm/s and decreasing gradually to zero, which is comparable to the experimental result, 1.5 mm/s to 0 as reported in Ref. [2]. From the above simulation results, one can also find that the deformation difference along the specimen originating from the prearranged disturbance is not amplified until a critical strain is reached, when serrations emerge in the load curves and the strain curves of different sections and diverge from one another, namely, FIGURE 5. Steplike strain curves of three adjacent sections. The results were obtained by assuming a random perturbation of the initial internal stress and with the nominal strain rate of s 1 International Journal for Multiscale Computational Engineering

9 MULTISCALE ANALYSIS AND NUMERICAL MODELING 235 unstable plastic deformation occurs in the specimen. It is therefore confirmed that the present model can reproduce the experimental fact that PLC effect takes place only after the deformation reaches some critical value. This feature can also be found in Fig. 6, which clearly depicts the relationship between jerky flow and dynamic strain aging. When the initial disturbance is introduced, there are several oscillations in the amplitude of the stress component σ dsa. Dominated by Eq. (4), these oscillations are rapidly damped, which indicates that the pinning and unpinning effects between mobile dislocations and solute atoms are in dynamic equilibrium. This balance is disrupted when the strain reaches the critical value and can no longer be kept during the subsequent deformation, and meanwhile the strain rate curve becomes comblike, i.e., jerky flow takes place. With increasing plastic strain, the average waiting time of mobile dislocations at their obstacles becomes longer due to the increasing impediment strength, which permits more solutes diffusing toward them, and therefore the pinning effect will temporarily take the advantage. This is the reason why the amplitudes of σ dsa show a gradually ascending tendency with increasing deformation. However, as described in Eq. (4), the bigger the stress σ dsa is, the stronger the unpinning effect. Consequently, the unstable flow becomes more violent and the comb teeth on the strain rate curve grow higher. In other words, the localized deformation on macroscopic scale associated with the PLC effect can be directly traced back to the pinning and unpinning processes on the microscopic scale, and only these two conflicting effects are considered here in a unified scheme. FIGURE 6. Strain rate and stress component σ dsa in one section. The results were obtained by assuming a random perturbation of the initial internal stress and with the nominal strain rate of s 1 Volume 3, Number 2, 2005

10 236 CHEN ET AL 4. CONCLUSIONS The PLC instability manifests the macroscopic deformation response of the alloys to the dislocation-dislocation interactions and the dislocation-solute interactions on lower length scales at certain deformation conditions, by which the dynamic strain aging may take place. When the research emphasis is put on the macroscopic behavior and how to control the phenomenon, these multiscale interactions should be considered in a collective way. The PLC effect is examined and simulated through a phenomenological model in the present work, based on a multiscale analysis. In this model, a new component of flow stress is introduced to characterize the competition of age hardening and strain-induced softening due to the collective interactions between solute atoms and mobile dislocations in the regime of dynamic strain aging. In other words, the microscopic pinning and unpinning processes are taken into account in an integrative scheme. It can satisfy the negative strain rate sensitivity of flow stress that has been well accepted as one necessary condition for the occurring of serrated yielding. A nonuniform spatial distribution of some model parameters was assumed in the numerical simulations, including a linear change of the initial cross-sectional area and a random perturbation of the initial internal stress, to fit the heterogeneous nature of the jerky flow. The modeling results, calculated for tensile tests using this heterogeneous model, are found in good agreement with the experimental serrated yielding and localized deformation curves. Comparing the oscillations on the curves of the strain rate and those on the curves of the stress component σ dsa, it is also indicated that the localized deformation behavior on the macroscopic scale associated with the PLC effect can be clearly related to the pinning and unpinning processes at the microscopic level. ACKNOWLEDGMENT This work was supported by the National Natural Science Foundation of China under Grants No and No Z. J. Chen is greatly indebted to Professor Yuanming Xia for his valuable suggestions. We would also like to thank the Lab of Mechanical and Material Science of USTC for their help in the experiments on MTS. REFERENCES 1. Pink, E., Kumar, S., and Tian, B., Serrated flow of aluminium alloys influenced by precipitates, Mater. Sci. Eng. A 280:17 24, Zhang, Q. C., Toyooka, S., Meng, Z. B., and Suprapedi, Investigation of slipband propagation in aluminum alloy with dynamic speckle interferometry, Proc. of SPIE 3585: , Zhang, Q. C., Toyooka, S., and Wu, X. P., Study of propagation and pulsation of slip band using dynamic digital speckle pattern interferometry, Proc. of SPIE 4537:69 75, Ma, E., Nanocrystalline materials: Controlling plastic instability, Nature Mat. 2:7 8, He, G., Eckert, J., Loeser, W., and Schultz, L., Novel Ti-base nanostructure-dendrite composite with enhanced plasticity, Nature Mater. 2:33 37, Chihab, K., Estrin, Y., Kubin, L. P., and Vergnol, J., The kinetics of the Portevin-Le Chatelier bands in an Al-5at%Mg alloy, Scripta Metall. 21: , Mertens, F., Franklin S. V., and Marder M., Dynamics of plastic deformation fronts in an aluminum alloy, Phys. Rev. Lett. 78: , Cottrell, A. H., A note on the Portevin-Le Chatelier, Phil. Mag. 44: , McCormick, P. G., A model for the Portevin- Le Chatelier effect in substitutional alloys, Acta Metall. 20: , McCormick, P. G., Theory of flow localisation due to dynamic strain aging, Acta Metall. International Journal for Multiscale Computational Engineering

11 MULTISCALE ANALYSIS AND NUMERICAL MODELING : , Van den Beukel A., Theory of the effect of dynamic strain aging on mechanical properties, Phys. Stat. Sol. A 30: , Estrin, Y., and McCormick, P. G., Modelling the transient flow behaviour of dynamic strain aging materials, Acta Metall. Mater. 39: , Kubin, L. P., and Estrin, Y., Evolution of dislocation densities and the critical conditions for the Portevin-Le Chatelier effect, Acta Metall. Mater. 38: , Estrin, Y., and Kubin, L. P., Special coupling and propagative plastic instabilities, in Continuum Models for Materials with Microstructure, H.- B. Muhlhaus (ed.), Wiley, New York, , Hahner, P., On the physics of the Portevin-Le Chatelier effect, Mater. Sci. Eng. A 207: , Hahner, P., Ziegenbein, A., Rizzi, E., and Neuhauser, H., Spatiotemporal analysis of Portevin-Le Chatelier deformation bands: Theory, simulation, and experiment, Phys. Rev. B 65:134109, Zhang, S., McCormick, P. G., and Estrin, Y., The morphology of Portevin-Le Chatelier bands: finite element simulation for Al-Mg-Si, Acta Mater. 49: , Kok, S., Beaudoin, A. J., Tortorelli, D. A., and Lebyodkin, M., A finite element model for the Portevin Le Chatelier effect based on polycrystal plasticity, Modelling Simul. Mater. Sci. Eng. 10: , Kok, S., Bharathi, M. S., Beaudoin, A. J., Fressengeas, C., Ananthakrishna, G., Kubin, L. P., and Lebyodkin, M., Spatial coupling in jerky flow using polycrystal plasticity, Acta Mater. 51: , Oden, J. T., Vemaganti, K., and Moes N., Hierarchical modeling of heterogeneous solids, Comput. Methods Appl. Mech. Eng. 172:3 25, Curtin, W. A., and Miller, R. E., Atomistic/ continuum coupling in computational materials science, Modelling Simul. Mater. Sci. Eng. 11:R33 R68, Diaz de la Rubia, T., Zbib, H. M., Khraishi, T. A., Wirth, B. D., Victoria, M., and Caturla, M. J., Multiscale modelling of plastic flow localization in irradiated materials, Nature 406: , Zbib, H. M., Diaz de la Rubia, T., and Bulatov, V., A multiscale model of plasticity based on discrete dislocation dynamics, J. Eng. Mater. Tech. 124:78 87, Onodera, R., Morikawa, T., and Higashida, K., Computer simulation of Portevin-Le Chatelier effect based on strain softening model, Mater. Sci. Eng. A : , Chen, Z. J., Zhang, Q. C., Jiang, Z. Y., Jiang, H. F., and Wu, X. P., A macroscopic model for the Portevin-Le Chatelier effect, J. Mater. Sci. Technol. 20(5): , Penning, P., Mathematics of the Portevin-Le Chatelier effect, Acta Metall. 20: , Chen, Z. J., Zhang, Q. C., Xiang, G. F., Jiang, H. F., Jiang, Z. Y., and Wu, X. P., Investigation on the dislocation-solute interaction during dynamic strain aging, Chin. Phys. Lett, (submitted). 28. Chen, Z. J., Zhang, Q. C., Dong, F. L., and Wu, X. P., Numerical simulation of the Portevin-Le Chatelier effect based on a heterogeneous model, in Proc. of Int. Conf. on Heterogeneous Materials Mechanics (ICHMM-2004), J. Fan, et al. (eds.), Chongqing University Press, Chongqing, 51 54, Tomita, Y., and Uchida, M., Computational modelling of micro- to macroscopic deformation behavior of amorphous polymer. in Proc. of Int. Conf. on Heterogeneous Materials Mechanics (ICHMM-2004). J. Fan, et al. (eds.), Chongqing University Press, Chongqing, 59 63, Volume 3, Number 2, 2005

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